Let G be a finite group. The prime graph of G is the graph Γ(G) whose set of vertices is the set Π(G) of all prime divisors of the order |G| and two different vertices p and q of which are connected by an edge if G has an element of order pq. We prove that if S is one of the simple groups L 5(4) and U 4(4) and G is a finite group with Γ(G) = Γ(S), then G has a normal subgroup N such that Π(N) ⊆ {2, 3, 5} and \( \frac{G}{N} \cong S \).
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References
M. Aschbacher and G. M. Seitz, “Involutions in Chevalley groups over fields of even order,” Nagoya Math. J., 63, No. 1, 1–91 (1976).
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford (1985).
M. R. Darafsheh, “Order of elements in the groups related to the general linear groups,” Finite Fields Appl., 11, 738–747 (2005).
M. R. Darafsheh, “Pure characterization of the projective special linear groups,” Ital. J. Pure Appl. Math., No. 23, 229–244 (2008).
M. R. Darafsheh and N. S. Karamzadeh, “On recognition property of some projective special linear groups by their element orders,” Util. Math., 75, 125–137 (2008).
M. Hagie, “The prime graph of a sporadic simple group,” Commun. Algebra, 31, No. 9, 4405–4424 (2003).
G. Higman, “Finite groups in which every element has prime power order,” J. London Math. Soc., 32, 335–342 (1957).
B. Khosravi, “Quasirecognition by prime graph of L 10(2),” Sib. Math. J., 50, No. 2, 355–359 (2009).
Behrooz Khosravi, Bahman Khosravi, and Behnam Khosravi, “A characterization of the finite simple group L 16(2) by its prime graph,” Manuscr. Math., 126, 49–58 (2008).
A. S. Kondratiev, “On prime graph components for finite simple groups,” Mat. Sb., 180, No. 6, 787–797 (1989).
V. D. Mazurov, M. C. Xu, and H. P. Cao, “Recognition of finite simple groups L 3(2m) and U 3(2m) by their element orders,” Alg. Logic, 39, No. 5, 567–585 (2000).
V. D. Mazurov, “Recognition of finite simple groups S 4(q) by their element orders,” Alg. Logic, 41, No. 2, 93–110 (2002).
V. D. Mazurov and G. Y. Chen, “Recognizability of finite simple groups L 4(2m) and U 4(2m) by spectrum,” Alg. Logic, 47, No. 1, 49–55 (2008).
V. D. Mazurov, “Characterization of finite groups by sets of element orders,” Alg. Logic, 36, No. 1, 23–32 (1997).
D. S. Passman, Permutation Groups, Benjamin, New York (1968).
W. J. Shi and W. Z. Yang, “A new characterization of A5 and finite groups in which every nonidentity element has prime order,” J. Southwest China Teachers Coll., 9–36 (1984).
A. V. Vasil’ev, “On connection between the structure of a finite group and the properties of its prime graph,” Sib. Math. J., 46, No. 3, 396–404.
A. V. Vasil’ev and M. A. Grechkoseeva, “On recognition by spectrum of finite simple linear groups over fields of characteristic 2,” Sib. Math. J., 46, No. 4, 593–600 (2005).
J. S. Williams, “Prime graph components of finite groups,” J. Algebra, 69, No. 2, 487–513 (1981).
A.V. Zavarnitsine, “Recognition of finite groups by the prime graph,” Alg. Logic, 45, No. 4 (2006).
A. V. Zavarnitsine, “Finite simple groups with narrow prime spectrum,” Sib. Electron. Math. Rep., 6, 1–12 (http://semr.math.nsc.ru/v6/p1--12.pdf) (2009).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 2, pp. 210–217, February, 2012.
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Nosratpour, P., Darafsheh, M.R. Recognition of the groups L 5(4) and U 4(4) by the prime graph. Ukr Math J 64, 238–246 (2012). https://doi.org/10.1007/s11253-012-0641-1
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DOI: https://doi.org/10.1007/s11253-012-0641-1